Jul 19, 1985 - such that $\varpi_{\gamma}\circ\phi*$. $=\tilde{\phi}_{*}\circ\varpi_{\gamma}$ on $TM$. We define a map $\phi:P_{\gamma}arrow P_{\gamma}$.
J. Math. Soc. Japan Vol. 39, No. 1, 1987
Contact structures on twistor spaces By Takashi NITTA and Masaru TAKEUCHI (Received July 19, 1985)
Introduction. A quaternionic K\"ahler manifold $(M, g, H)$ is a Riemannian manifold $(M, g)$ together with a coefficient bundle $H$ of quaternions. The twistor space $Z$ of $(M, g, H)$ , which is a complex manifold fibring over $M$, has a natural complex contact structure 7 and a natural Einstein pseudo-K\"ahler metric , provided that $(M, g)$ has non-zero scalar curvature (Salamon [8]). In this note we shall study the automorphism groups of these structures. denote the group of automorphisms of Let $Aut(M, g, H)$ and $(M, g, H)$ and the one of isometric contact automorphisms of respectively. Then each element in $Aut(M, g, H)$ can be lifted to an element in in a natural way. We show first that the lifting homomorPhism $Aut(M, g, H)arrow Aut(Z, \gamma,\overline{g})$ is an isomorphjsm (Theorem 3.1). be the Lie algebra of infinitesimal contact autoLet and morphisms of and the one of infinitesimal isometric contact automorphisms respectively. We prove next that of is the complexification of (Corollary 2 to Theorem 3.2). This may be viewed as an analogue to the theorem of Matsushima to the effect that the Lie algebra of holomorphic vector fields on a compact Einstein K\"ahler manifold is the complexification of the Lie algebra of Killing vector fields. Lastly we study a certain uniqueness of quaternionic K\"ahler structures. We prove: SuPpose that a compact complex contact manifold $M$ admits a Kahler metric and has the vamshing first integral homology. Then a complex contact structure on $M$ is unique up to automorphisms of $M$ (Theorem 1.7). Making use of this and previous results we show the following uniqueness: Let compact quaternionic Kahler manifolds $(M, g, H)$ and $(M’, g’, H’)$ with positive scalar curvatures have the same twistor sPace Z. SuppOse that $Z$ is a kahlerian -sPace of Boothby type (see \S 1 for the definition). Then $(M, g, H)$ and $(M’, g’, H’)$ are equivalent to each other (Theorem 4.2). Note that all the known examples of twistor spaces of compact quaternionic K\"ahler manifolds with positive scalar curvature are k\"ahlerian C-spaces of Boothby type. The first named author would like to express his hearty thanks to Professors $\overline{g}$
$Aut(Z, \gamma,\overline{g})$
$(Z, \gamma,\overline{g})$
$Aut(Z, \gamma,\overline{g})$
$\mathfrak{a}(Z, \gamma)$
$\mathfrak{a}(Z, \gamma,\overline{g})$
$(Z, \gamma)$
$(Z, \gamma,\overline{g})$
$\mathfrak{a}(Z, \gamma)$
$\mathfrak{a}(Z, \gamma,\overline{g})$
$C$
140
T. NITTA and M. TAKEUCHI
H. Ozeki, T. Mabuchi and Y. Sakane for their constant encouragement and instruction.
\S 1. Complex contact structures. In this section we recall the basic results on complex contact structures and prove a uniqueness theorem for complex contact structures on certain k\"ahlerian manifolds. Let $M$ be a (connected) complex manifold of odd dimension $2n+1$ . By a system of contact forms on $M$ we mean an open cover of $M$ together with a system of holomorphic l-forms such that $\{U_{i}\}$
$\{\gamma_{i}\}$
is defined on
(1.1)
each
(1.2)
we have
$U_{i}\cap U_{j}$
$\gamma_{i}$
$\gamma_{i}=f_{ij}\gamma_{j}$
on
$U_{i}$
and
$U_{i}\cap U_{j}$
$7i^{\wedge(dr_{i})^{n}\neq 0}$
,
where
$f_{ij}$
everywhere
on
$U_{i}$
; and
is a holomorphic function on
.
Two systems of contact forms and are said to define the same complex contact structure if , where on is a holomorphic function on of a complex manifold $M$ and a complex . A pair contact structure on $M$ is called a complex contact manifold. Given a complex , we define a holocontact structure on $M$ defined by contact forms morphic subbundle of the (holomorphic) tangent bundle $TM$ by $\{U_{i}, \gamma_{i}\}$
$\gamma_{i}=h_{i\lambda}\delta_{\lambda}$
$U_{i}\cap V_{\lambda}$
$\{V_{\lambda}, \delta_{\lambda}\}$
$U_{i}\cap V_{\lambda}$
$h_{i\lambda}$
$(M, \gamma)$
$\gamma$
$\{U_{i}, \gamma_{i}\}$
$\gamma$
$E_{\gamma}$
$(E_{\gamma})_{x}=\{X\in T_{x}M ; \gamma_{i}(X)=0\}$
Let be the quotient line bundle and jection, so that we have an exact sequence $L_{\gamma}=TM/E_{\gamma}$
if $\varpi_{\gamma}$
:
$x\in U_{i}$
.
$TMarrow L_{\gamma}$
the natural pro-
$0arrow E_{\gamma}arrow TMarrow L_{\gamma}\varpi_{\gamma}arrow 0$
(1.3)
of holomorphic vector bundles over $M$. Let and be complex contact manifolds. By a contact isomorphism $\phi:(M, \gamma)arrow(M’, \gamma’)$ we mean a holomorphic diffeomorphism $\phi:Marrow M’$ $TMarrow TM’$ induces an isomorphism such that the differential . For a complex contact manifold a contact isomorphism of onto itself is called a contact automorphism. Denote by $Aut(M, \gamma)$ the group of all contact automorphisms of . It is a complex Lie group if $M$ is compact (Boothby [4]). A holomorphic vector field on $M$ is called an infinitesimal contact automorphism of if it generates a local flow of (local) contact automorphisms of denote the complex Lie algebra of all infinitesimal . Let . If $M$ is compact, contact automorphisms of is identified with Lie $Aut(M, \gamma)$ , the Lie algebra of $Aut(M, \gamma)$ . $(M’, \gamma’)$
$(M, \gamma)$
$\phi_{*}:$
$E_{\gamma}arrow E_{\gamma’}$
$(M, \gamma)$
$(M, \gamma)$
$(M, \gamma)$
$(M, \gamma)$
$(M, \gamma)$
$\alpha(M, \gamma)$
$(M, \gamma)$
$\mathfrak{a}(M, \gamma)$
Twistor spaces
141
In what follows we recall some basic results on complex contact structures be a complex contact manifold Let of dimension $2n+1$ with defined by contact forms . (cf. Kobayashi [6], [7], Boothby [4]).
$(M, \gamma)$
$\{U_{i}, \gamma_{i}\}$
$\gamma$
THEOREM 1.1. The system holomorphic pairing:
induces a non-degenerate alternating
$\{U_{i}, d\gamma_{i}\}$
.
$E_{\gamma}\cross E_{\gamma}arrow L_{\gamma}$
THEOREM 1.2. The canonical line bundle $K_{M}=\wedge^{2n+1}T^{*}M$ of morphically isomorphic to . In particular, first Cherrt classes $L_{\overline{\gamma}}^{(n+1)}$
$c_{1}(M)=(n+1)c_{1}(L_{\gamma})$
is holosatisfy
$M$ $c_{1}$
.
be the holomorphic $c*$ -bundle associated to Let : the right action of $a\in C^{*}$ on bundle of frames of , and denote by by on define a holomorphic l-form $\hat{\pi}$
$P_{\gamma}arrow M$
$L_{\gamma},$
$R_{a}$
$L_{\gamma}$
for
$\theta_{\gamma}(X)=u^{-1}\varpi_{\gamma}(\hat{\pi}_{*}X)$
THEOREM 1.3.
is semi-basic, vector of $P$ ; (1.4)
$\theta$
(1.5)
$R_{a}^{*}\theta=a^{-1}\theta$
(1.6)
$\Theta=d\theta$
l-form on
We write
for
$i$
.
$\theta$
$e.,$
each
$P,$
$\theta$
$P_{\gamma}$
. We
for
$P_{\gamma},$
$X\in T{}_{u}P_{\gamma},$
put
$\theta_{\gamma}$
$\Theta_{\gamma}=d\theta_{\gamma}$
$u\in P_{\gamma}$
$e.$
$P_{\gamma}$
, the
. We
,
.
. We have then
is annihilated by the contraction
$a\in C^{*};$
is a symplectic
.
$P_{\gamma}$
$\theta_{\gamma}$
and call it the canonical
$i$
of any
vertical
and
2-form on . $P$
Now we fix a complex manifold $M$ of odd dimension and denote by $C(M)$ the set of all complex contact structures on $M$. Note that the group $Aut(M)$ of all holomorphic automorphisms of $M$ acts on $C(M)$ in a natural way. Next let us of holomorphic $c*$ -bundle over $M$ and a holomorphic consider a pair on $P$ satisfying (1.4), (1.5) and (1.6). Two such pairs and l-form are said to be equivalent if there exists a holomorphic $c*$ -bundle isomorphism $\Phi:Parrow P’$ inducing the identity on $M$ such that . The set equivalence pairs . classes of such is denoted by of all $P$
$(P, \theta)$
$\theta$
$(P, \theta)$
$(P’, \theta’)$
$\Phi^{*}\theta’=\theta$
$\mathcal{P}(M)$
THEOREM 1.4. The correspondence
$\gamma->(P_{\gamma}, \theta_{\gamma})$
induces a bijection
Let be a complex contact manifold. For each uniquely of a holomorphic bundle automorphism exists by on $TM$. We define a map $(M, \gamma)$
$\phi_{*}$
$=\tilde{\phi}_{*}\circ\varpi_{\gamma}$
$\phi:P_{\gamma}arrow P_{\gamma}$
$\hat{\phi}(u)=\phi_{*}\circ u$
for
$u\in P_{\gamma}$
,
$C(M)arrow \mathcal{P}(M)$
$\phi\in Aut(M, \gamma)$ $L_{\gamma}$
such that
.
there $\varpi_{\gamma}\circ\phi*$
142
T. NITTA and M. TAKEUCHI
which is a holomorphic $c*$ -bundle automorphism of such that by prolongation the group of all of . We denote This is called the (not necessarily inducing the holomorphic $c*$ -bundle automorphisms of $M$ ) with identity on be the complex Lie . Moreover let algebra of all holomorphic vector fields $X$ on such that the local flow , which is the same as that generated by $X$ is contained (locally) in $a\in C^{*}$ $X$ $R_{a*}X=X$ for each , where and denotes the Lie derivation. is a complex Lie group with Lie If $M$ is compact, , and . For each $\phi\in Aut(M, \gamma)$ the prolongation belongs to , which is called the hence we have a homomorphism , which is prolongation. This gives rise to a homomorphism prolongation. called the infinitesimal $\hat{\pi}\circ\hat{\phi}=\phi\circ_{l\vee}^{\wedge}\sim$
$P_{\gamma}$
$Aut(P_{\gamma}, \theta_{\gamma})$
$\phi$
$\Phi$
$P_{\gamma}$
$\Phi^{*}\theta_{\gamma}=\theta_{\gamma}$
$\mathfrak{a}(P_{\gamma}, \theta_{\gamma})$
$P_{\gamma}$
$Aut(P_{\gamma}, \theta_{\gamma})$
$\mathcal{L}_{X}\theta_{\gamma}=0$
$Aut(P_{\gamma}, \theta_{\gamma})=$
$Aut(P_{\gamma}, \theta_{\gamma})$
$\phi$
$\mathfrak{a}(P_{\gamma}, \theta_{\gamma})$
$Aut(P_{\gamma}, \theta_{\gamma})$
$Aut(M, \gamma)arrow^{\wedge}Aut(P_{\gamma}, \theta_{\gamma})$
$\mathfrak{a}(M, \gamma)arrow^{\wedge}\mathfrak{a}(P_{\gamma}, \theta_{\gamma})$
THEOREM 1.5. The pr0l0ngati0n which is a complex Lie isomorphism if $M$ is compact. is also an isomorphism. pr0l0ngati0n
is an isomorphjsm, Therefore the infinitesimal
$Aut(M, \gamma)arrow^{\wedge}Aut(P_{\gamma}, \theta_{\gamma})$
$\mathfrak{a}(M, \gamma)arrow \mathfrak{a}(P_{\gamma}, \theta_{\gamma})$
THEOREM 1.6. Let : define a linear map $\varpi_{\gamma}$
$\Gamma(L_{\gamma})$
be the
space of
$\mathfrak{a}(M, \gamma)arrow\Gamma(L_{\gamma})$
$\varpi_{\gamma}$
$L_{\gamma}$
, and
for
$X\in \mathfrak{a}(M, \gamma),$
$x\in M$
.
is an isomorphism.
COROLLARY. such that
Let
$F(P_{\gamma})$
be the
$\sigma(u\cdot a)=a^{-1}\sigma(u)$
For each
of
by
$\varpi_{\gamma}(X)(x)=\varpi_{\gamma}(X_{x})$
Then
all holomorphic sections
$\sigma\in F(P_{\gamma})$
space of
all holomorphjc
for each
a holomorphjc vector field
$\iota(X)$
on
$X$
$\iota(X)\Theta_{\gamma}+d\sigma=0$
where denotes the contraction by X. linear isomorphism .
$u\in P_{\gamma},$
$P_{\gamma}$
functions
$a\in C^{*}$
$\sigma$
on
$P_{\gamma}$
.
is uniquely determined by
,
Then the corresp0ndence
$\sigma-,X$
gives a
$F(P_{\gamma})arrow \mathfrak{a}(P_{\gamma}, \theta_{\gamma})$
For each the holomorphic function on defined by is a function in , and the correspondence gives a linear isomorphism . Therefore the corollary follows from Theorems 1.5, 1.6 and the familiar identity $\iota(X)\circ d+d\circ\iota(X)=X_{X}$ . . . . PROOF.
$s\in\Gamma(L_{\gamma})$
$\sigma(u)=u^{-1}s(\hat{\pi}(u))$
$\sigma$
$F(P_{\gamma})$
$P_{\gamma}$
$s-,\sigma$
$\Gamma(L_{\gamma})arrow F(P_{\gamma})$
$q$
$e$
$d$
A complex contact manifold is said to be homogeneous if $Aut(M, \gamma)$ acts transitively on $M$. A complex manifold is said to be kahlerian if it admits a K\"ahIer metric. $(M, \gamma)$
EXAMPLE 1.1 (Boothby [4], [5]). Let
$\mathfrak{g}$
be a complex simple Lie algebra.
Twistor spaces
143
of and identify the root system of with a Take a Cartan subalgebra of . Let of by means of the Killing form subset of the real part and put be the highest root with respect to a linear order on the -eigenspace of $ad(H_{0})$ in , we define a . Denoting by by subalgebra of $\mathfrak{h}$
$\mathfrak{g}$
$\mathfrak{h}_{R}$
$\mathfrak{g}$
$(, )$
$\mathfrak{h}$
$a_{0}\in \mathfrak{h}_{R}$
$\mathfrak{g}$
$\mathfrak{h}_{R}$
$H_{0}=(2/(\alpha_{0}, \alpha_{0}))\alpha_{0}$
$\lambda$
$\mathfrak{g}$
$\mathfrak{g}_{\lambda}$
$\mathfrak{u}$
$\mathfrak{g}$
$\mathfrak{u}=\mathfrak{g}_{0}+\mathfrak{g}_{1}+\mathfrak{g}_{2}$
Let Lie
$G$
.
be the connected complex Lie group with the trivial center such that , and $U$ the normalizer of 11 in . Then the quotient complex manifold $G$
$G=\mathfrak{g}$
$M=G/U$ is compact, simply connected and k\"ahlerian. It is called a kahlerian -space of . Boothby type. It is shown that $M$ has always odd dimension and that on by with and define a linear form Choose $C$
$\mathfrak{g}_{2}\neq\{0\}$
$E\neq 0$
$E\in \mathfrak{g}_{2}$
$\theta^{*}$
$\mathfrak{g}$
for
$\theta^{*}(X)=(E, X)$
$X\in \mathfrak{g}$
.
Put $U_{0}=\{a\in G ; Ad(a)E=E\}$ .
Then is a normal subgroup of $U$ and the quotient group with $c*$ . Thus the quotient complex manifold $U_{0}$
$U/U_{0}$
is identified
$P=G/U_{0}$
has a natural structure of a holomorphic $c*$ -bundle over $M$. And there exists being on $P$ whose pull back to is equal to a holomorphic l-form regarded as a left G-invariant l-form on . Furthermore it is verified that determines a complex satisfies (1.4), (1.5) and (1.6). Thus by Theorem 1.4 contact structure on $M$. Our correspondence $G$
$\theta$
$\theta^{*},$
$\theta^{*}$
$G$
$\theta$
$\theta$
$\gamma$
$\mathfrak{g}-\geq(M, \gamma)$
induces a bijection from the set of all isomorphism classes of complex simple Lie algebras onto the set of all contact isomorphism classes of compact simply connected homogeneous complex contact manifolds. Actually we have Lie $Aut(M, \gamma)$ in the above construction. $\mathfrak{g}=$
A complex contact structure on a k\"ahlerian C-space of Boothby type is essentially unique, because we have the following theorem.
THEOREM 1.7. Let that $M$ is kahlerian and
$(M, \gamma_{0})$
be a compact complex contact manifold. Supp0se . Then $Aut(M)$ acts transitively on $C(M)$ .
$H_{1}(M, Z)=\{0\}$
PROOF. Let $h^{p.q}=\dim H^{q}(M, \Omega^{p})$ and Since $M$ is compact k\"ahlerian, we have
$b_{r}=the$
r-th Betti number of
.
$M$
144
T. NITTA and M. TAKEUCHI $b_{r}=$
In particular, by
$\sum_{P+q=r}h^{p,q}$
$H_{1}(M, Z)=\{0\}$
(1.7)
,
we get
$h^{p,q}=h^{q,p}$
$b_{1}=0$
$h^{0.1}=0$
.
and hence
.
We put $L=L_{\gamma_{0}}$
,
$P=P_{\gamma_{0}}$
.
Take an arbitrary $\gamma\in C(M)$ . By Theorem 1.2 we have where dim $M=2n+1$ . Therefore we have (1.8)
$(n+1)c_{1}(L_{\gamma})=(n+1)c_{1}(L)$
$(n+1)c_{1}(L_{\gamma})=c_{1}(M)$
,
.
On the other hand, the exact sequence $0arrow Zarrow Carrow c*arrow 0$
of abelian groups yields the cohomology exact sequence $c_{1}$
$H^{1}(M, O)arrow H^{1}(M, O^{*})arrow H^{2}(M, Z)$
where dim $H^{1}(M,
O)=h^{0,1}=0$
by (1.7).
Thus we get the exact sequence $c_{1}$
(1.9)
,
$0arrow H^{1}(M, O^{*})arrow H^{2}(M, Z)$
.
Moreover the assumption $H_{1}(M, Z)=\{0\}$ together with the universal coefficient theorem implies that $H^{2}(M, Z)$ has no torsion. Hence by (1.8) and (1.9) we have (1.10)
$L_{\gamma}\cong L$
Next let (1.5) and (1.6). quotient
$\hat{\mathcal{U}}$
for each
$\gamma\in C(M)$
.
on $P$ satisfying (1.4), be the set of all holomorphic l-forms , and the Then is invariant under the multiplication by $\theta$
$\hat{\mathcal{U}}$
$C^{*}$
$\mathcal{U}(M)=\hat{\mathcal{U}}/C^{*}$
is identified with $C(M)$ by Theorem 1.4, (1.10) and the compactness of $M$. We identify the space of all holomorphic l-forms on satisfying (1.4) and (1.5) , and with the space $\Gamma(M, T^{*}M\otimes L)$ of all holomorphic sections of thus . In the same way we identify on such that with the space of all semi-basic holomorphic $(2n+1)$ -forms $R_{a}^{*}a=a^{-(n+1)}a$ for each . We fix an isomorphism from the anticanonical (cf. Theorem 1.2). We have then onto line bundle $\theta$
$P$
$T^{*}M\otimes L$
$\hat{\mathcal{U}}\subset\Gamma(M, T^{*}M\otimes L)$
$\Gamma(M, \wedge^{2n+1}T^{*}M\otimes L^{n+1})$
$a$
$a\in C^{*}$
$K_{M}^{*}$
$L^{n+1}$
$\kappa$
$P$
Twistor
spaces
145
$\Gamma(M, \wedge^{2n+1}T^{*}M\otimes L^{n+1})=\Gamma(M, K_{M}\otimes L^{n+1})$
$arrow^{\cong}\Gamma(M, K_{M}\otimes K_{M}^{*})=\Gamma(M, 1)=C$
.
$\kappa^{*}$
Here Let
is given as follows.
$\kappa^{*}$
satisfy
$\xi_{0}\in(K_{M})_{x_{0}}$
$K_{M}$
. Then for
Fix a point $x_{0}\in M$ and with , where is the pairing between $v_{0}\in(K_{M}^{*})_{x_{0}}$
$\langle v_{0}, \xi_{0}\rangle=1$
$a\in\Gamma(\Lambda f, K_{M}\otimes L^{n+1})$
, regarded
$v_{0}\neq 0$
as a homomorphism
.
and
$K_{M}^{*}$
$\langle, \rangle$
$a;KMarrow L^{n+1}$ ,
we have $\kappa^{*}(\alpha)=\langle\kappa^{-1}(a(v_{0})), \xi_{0}\rangle$
Now we define a map
$F:\Gamma(M, T^{*}M\otimes L)arrow C$
.
by $\kappa^{*}$
$F(\theta)=\theta\wedge(d\theta)^{n}\in\Gamma(M, \wedge^{zn+1}T^{*}M\otimes L^{n+1})=C$
,
explicitly, by $F(\theta)=\langle\kappa^{-1}[(\theta\Lambda(d\theta)^{n})(v_{0})], \xi_{0}\rangle$
for
$\theta\in\Gamma(M, T^{*}M\otimes L)$
.
It is a homogeneous holomorphic function on $\Gamma(M, T^{*}M\otimes L)$ of degree $n+1$ if and only if is a symplectic 2-form on $P$. Therefore such that $F\neq 0$ (since ) and is given by $d\theta$
$F(\theta)\neq 0$
$\hat{\mathcal{U}}$
$F(\theta_{\gamma_{0}})\neq 0$
$\hat{\mathcal{U}}=\{\theta\in\Gamma(M, T^{*}M\otimes L) ; F(\theta)\neq 0\}$
Hence that
$\hat{\mathcal{U}}$
.
is a -invariant open connected subset of $\Gamma(M, T^{*}M\otimes L)$ . It follows is identified with an open connected subset of the projective space $C^{*}$
$\mathcal{U}(M)$
associated to $\Gamma(M, T^{*}M\otimes L)$ . Let $Aut(L)$ denote the complex (not necessarily Lie group of all holomorphic bundle automorphisms of inducing the identity on $M$ ). Then we have an exact sequence $P(\Gamma(M, T^{*}M\otimes L))$
$L$
$1arrow c*arrow Aut(L)arrow^{\rho}Aut(M)arrow 1$
is the natural homomorphism. Here the surof complex Lie groups, where jectivity of follows from (1.10). For each $g\in Aut(M)$ we choose an element $s_{g}\in Aut(L)$ with $\rho(s_{g})=g$ . Then the tensor product of the natural action of on $T^{*}M$ with on $\Gamma(M, T^{*}M\otimes L)$ , and hence it induces a linear action of $P(\Gamma(M, T^{*}M\otimes L))$ , which leaves on induces a projective action of $C(M)$ , the action of with invariant. Under our identification of on defined in the above way corresponds to the natural action of on $C(M)$ . is holomorphic, since the Note that our action of $Aut(M)$ on bundle : $Aut(L)arrow Aut(M)$ has a local holomorphic section $g-s_{g}$ around each point of $Aut(M)$ . Therefore it suffices to show the transitivity of $Aut(M)$ on $\rho$
$\rho$
$g$
$g$
$s_{g}$
$\mathcal{U}(M)$
$g$
$\mathcal{U}(M)$
$g$
$\mathcal{U}(M)$
$g$
$\mathcal{U}(M)$
$\rho$
$\mathcal{U}(M)$
.
be arbitrary. Dualizing the exact sequence (1.3) and tensoring , we get an exact sequence
Let
$L_{\gamma}$
$\gamma\in C(M)$
T. NITTA and M. TAKEUCHI
146
$0arrow 1arrow T^{*}M\otimes L_{\gamma}arrow E_{\gamma}^{*}\otimes L_{\gamma}arrow 0$
.
In the associated cohomology exact sequence ,
$0arrow\Gamma(M, 1)arrow\Gamma(M, T^{*}M\otimes L_{\gamma})arrow\Gamma(M, E_{\gamma}^{*}\otimes L_{\gamma})-arrow H^{1}(M, O)$
we have
$\Gamma(M, 1)=C$
dim
and
$H^{1}(M, O)=\{0\}$
Therefore
by (1.7).
$\Gamma(M, T^{*}M\otimes L_{\gamma})=\dim\Gamma(M, E_{\gamma}^{*}\otimes L_{\gamma})+1$
,
to the pairing . But, by tensoring and hence dim in Theorem 1.1, we get a non-degenerate holomorphic pairing . Therefore we have . Thus $\mathcal{U}(M)=\dim\Gamma(M, E_{\gamma}^{*}\otimes L_{\gamma})$
$L_{\gamma}^{-1}$
$E_{\gamma}\cross E_{\gamma}arrow L_{\gamma}$
$E_{\gamma}\cong(E_{\gamma}\otimes L_{\overline{\gamma}^{1}})^{*}=E_{\gamma}^{*}\otimes L_{\gamma}$
$E_{\gamma}\otimes(E_{\gamma}\otimes L_{\overline{\gamma}^{1}})arrow 1$
dim $\mathcal{U}(M)=\dim\Gamma(M, E_{\gamma})$ .
(1.11)
On the other hand, in the cohomology exact sequence $0arrow\Gamma(M, E_{\gamma})arrow\Gamma(M, TM)-\Gamma(M\varpi_{\gamma}L_{\gamma})$
associated to (1.3), we have $\Gamma(M, L_{\gamma})\cong LieAut(M, \gamma)$ and
and by Theorem 1.6 Hence, together with (1.11) we
$\Gamma(M, TM)\cong LieAut(M)$ ,
$\varpi_{\gamma}$
is surjective.
get
dim $\mathcal{U}(M)=\dim
Aut(M)$
–
$\dim Aut(M, \gamma)$
for each
$\gamma\in C(M)$
.
locally transitively at each point of It follows that $Aut(M)$ acts on . is connected, we obtain the transitivity of $Aut(M)$ on Since $\mathcal{U}(M)$
$\mathcal{U}(M)$
$q$
REMARK. be replaced by
.
$\mathcal{U}(M)$
$\mathcal{U}(M)$
. . . $e$
$d$
As is seen from the proof, the assumptions in the theorem may “ $H_{1}(M, Z)$ has no torsion and $H^{1}(M, \mathcal{O})=\{0\}$ .
\S 2. Quaternionic K\"ahler manifolds. In this section we give the definition of quaternionic K\"ahler manifolds and recall some basic results on them. We denote by $H$ the algebra of real quaternions and by the subspace of pure imaginary quaternions. Let $(M, g)$ be a Riemannian manifold of dimension $4n$ . A subalgebra bundle $H$ of the bundle End $(TM)$ of endomorphisms of the tangent bundle $TM$ is called a quaternionic Kahler structure on $(M, g)$ if $\mathcal{I}mH\subset H$
(2.1)
for each point
the restriction bundles;
$H|U$
there is an open set $U$ of $M$ with is isomorphic to the product bundle
$x\in M$
to
$U$
$x\in U$ $U\cross H$
such that as algebra
Twistor spaces
denoting by the subbundle of isomorphisms in (2.1), we have (2.2)
$H$
$\mathcal{I}_{m}H$
which corresponds to
for
$g(zX, Y)+g(X, zY)=0$
147
$X,$
$Y\in T_{x}M,$
$U\aleph \mathcal{I}mH$
$z\in(\mathcal{I}mH)_{x}$
under
;
and
is a parallel subbundle of End $(TM)$ with respect to the connection of $(M, g)$ . induced by the Riemannian connection (2.3)
$H$
$\nabla$
Given such a structure the form
,
$H$
the set
$P(H)$
of all orthonormal frames of
$\{e_{1}, Ie_{1}, Je_{1}, Ke_{1}, \cdots , e_{n}, Ie_{n}, Je_{n}, Ke_{n}\}$
$(M, g)$
of
,
with $I^{2}=J^{2}=-1,$ $IJ=-JI=K$, is a subwhere $e_{i}\in T_{x}M$ and bundle of the orthonormal frame bundle $O(M)$ with the structure group $Sp(n)Sp(1)\subset O(4n)$ . Here $Sp(n)Sp(1)$ is defined as follows. We identify with and denote by $\rho(Sp(1))$ the subgroup of $O(4n)$ of right multiplications by $Sp(1)=\{h\in H;|h|=1\}$ . Let $Sp(n)\subset O(4n)$ be the centralizer of $\rho(Sp(1))$ in $O(4n)$ and define $Sp(n)Sp(1)$ to be the product $Sp(n)p(Sp(1))$ in $0(4n)$ . Actually $Sp(n)Sp(1)\subset SO(4n)$ and we have an exact sequence $I,$
$J,$
$K\in(\mathcal{I}mH)_{x}$
$H^{n}$
$R^{4n}$
$1arrow Z_{2}arrow Sp(n)\cross sp(1)arrow Sp(n)Sp(1)arrow 1$
.
on $O(M)$ reduces to the connection on By (2.3) the Riemannian connection $P(H)$ , and $M$ has a natural orientation determined by the reduction $P(H)\subset O(M)$ . The triple $(M, g, H)$ , where $H$ is a quaternionic K\"ahler structure on $(M, g)$ , , and in case $n=1$ provided is called a quaternionic Kahler manifold in case that $(M, g)$ is Einstein and anti-selfdual in the sense of Atiyah-Hitchin-Singer [3] with respect to the natural orientation. It is known (Alekseevskii [1]) that for a quaternionic K\"ahler manifold $(M, g, H)$ , $(M, g)$ is Einstein also in the and it is irreducible in the case where the (constant) scalar curvature case . Let $(M, g, H)$ and $(M’, g’, H’)$ be quaternionic K\"ahler manifolds. By an isomorphism (resp. equivalence) $\psi:(M, g, H)arrow(M’, g’, H’)$ we mean an isometry (resp. homothety) $\psi:(M, g)arrow(M’, g’)$ such that . An automorPhism of $(M, g, H),$ $Aut(M, g, H)$ and are defined in the analogous way to $Aut(M, g, H)$ is a Lie group, since it is a closed contact structures. The group subgroup of the group $K(M, g)$ of all isometries of $(M, g)$ . If is complete, we have Lie $Aut(M, g, H)=a(M, g, H)$ . In the following in this section, $(M, g, H)$ will denote a quaternionic K\"ahler manifold of dimension $4n$ . We recall first another description of quaternionic K\"ahler structures by Salamon [8]. If a complex $Sp(n)\cross Sp(1)$ -module $V$ is given, we get a vector bundle over $M$ associated to $P(H)$ with the induced connection, which will be denoted by the corresponding boldface $V$. In general $V$ is locally defined, but $\nabla$
$\nabla$
$n\geqq 2$
$n\geqq 2$
$t\neq 0$
$\psi_{*}H\psi_{*}^{-1}=H’$
$\mathfrak{a}(M, g, H)$
$g$
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T. NITTA and M. TAKEUCHI
it is globally defined if, for example, the action of $Sp(n)\cross Sp(1)$ factors through $Sp(n)Sp(1)$ . If has an $Sp(n)\cross Sp(1)$ -invariant structure, it carries over to the fibres of $V$. For example, if has an $sp(n)\cross Sp(1)$ -invariant real structure $V$ . In this case the set of fixed has the induced real structure points of in is a real vector bundle over $M$, which will be denoted $V$
$V$
$vrightarrow\overline{v}$
$v->\overline{v},$
$vrightarrow\overline{v}$
by
$V_{R}$
$V$
.
Let $E$ be the standard complex $Sp(n)$ -module of dimension $2n$ . It has an $Sp(n)$ -invariant antilinear map and an $Sp(n)$ -invariant nonwith and degenerate alternating bilinear form such that is a hermitian inner product on . By a standard basis of that $E$ we with respect to \langle , such that mean a unitary basis $E$ by the map . We shall often identify with $\approx v=-v$
$vrightarrow\tilde{v}$
$\omega_{E}(\tilde{u},\tilde{v})=\overline{\omega_{E}(u,v)}$
$\omega_{E}\in\wedge^{2}E^{*}$
$E$
$\langle\langle u, v\rangle\rangle=\omega_{E}(u,\tilde{v})$
$\{e_{1}, \cdots , e_{2n}\}$
$\langle$
$\tilde{e}_{i}=e_{n+i}$
$\rangle\rangle$
$E^{*}$
$(1\leqq i\leqq n)$
(2.4)
$v-\geq\omega_{E}(\cdot, v)$
.
Under this identification we have (2.5)
for a standard basis
$\omega_{E}=\sum_{i=1}^{n}e_{i}\wedge\tilde{e}_{i}$
In case ’rhen We put
$n=1$
$E\otimes F$
$\{e_{i}\}$
.
as $sp(n)\cross Sp(1)$ -modules. and we write for . We regard . has an $sp(n)\cross Sp(1)$ -invariant real structure defined by $E$
$F$
$F$
$E$
$\overline{e\otimes h}=\tilde{e}\otimes\tilde{h}$
$\tau*=\{v\in E\otimes F ; \overline{v}=v\}$
,
$g=\omega_{E}\mathfrak{U}_{F}\in S^{2}(E\otimes F)=S^{2}((T^{*})^{C})$
,
denotes the symmetric product $V\vee V$ . Then defines an $Sp(n)\cross Sp(1)-$ , and we have a global identification invariant inner product on
where
$S^{2}V$
$g$
$\tau*$
$T^{*}M=(E\otimes F)_{R}$
including the metrics. Next we put by the composition on
$T^{c}=E^{*}\otimes F^{*}$
and define an action of
$S^{2}F$
$T^{c}$
(2.6)
$S^{2}F\otimes T^{C}\subsetarrow(F\otimes F)\otimes(E^{*}\otimes F^{*})arrow(F\underline{\otimes F^{*})\otimes(E^{*}\otimes F}^{*})$
$arrow E^{*}\otimes F^{*}=T^{c}$
of the identification (2.4) and the indicated contraction. We denote by the real forms of and with respect to the real structures defined and by . Then leaves invariant and we have a global identification
$(S^{2}F)_{R}$
$T$
$vrightarrow\tilde{v}$
$T^{c}$
$S^{2}F$
$(S^{2}F)_{R}$
$T$
(2.7)
as subbundles of End $(TM)$ . We define the twistor space
$\mathcal{I}mH=(S^{2}F)_{R}$
$Z$
of
$(M, g, H)$
by
Twistor spaces
149
;
$Z=\{z\in H ; z^{2}=-1\}=\iota_{Z\in \mathcal{I}mH}$
$|z|^{2}=4n$
},
where . means the fibre norm of End $(TM)$ defined by . Denote by $q:Zarrow M$ is an the projection induced by the one of End $(TM)$ . The twistor space $M$ -bundle over with the induced connection. Also it is described as follows. Let $F_{0}=F-$ { $zero$ section}, $g$
$|$
$Z$
$S^{2}$
and denote by $\pi;F_{0}arrow Z$ with
$p;F_{0}arrow M$ $q\circ\pi=p$
the projection. We may define a (locally defined) map
by
$\pi(h)=\frac{1}{\omega_{F}(h,\tilde{h})}\sqrt{-1}h\vee\tilde{h}$
for
$h\in F_{0}$
,
under the identification (2.7). We have $\pi(h)=\pi(h’)$ if and only if there exists with $h’=h\cdot a$ , and therefore $Z$ is identified with the projective bundle $P(F)=F_{0}/c*$ associated to $F$ , which is a globally defined manifold. Under this identification, $Z_{x}=(F_{0})_{x}/C^{*}=P(F_{x})\cong P_{1}(C)$ . Note that $\pi;F_{0}arrow Z$ is a (locally defined) smooth $c*$ -bundle. We define a complex structure on $Z$ in the following way. Decompose as the sum $a\in C^{*}$
$TF_{0}$
$TF_{0}=\hat{\mathcal{H}}\oplus\hat{\mathcal{V}}$
of the horizontal distribution connection. For with
for the induced and the vertical distribution $T_{x}M$ through $x=p(h)$ , isomorphic to is and corre. Getting together the complex structure on , we sponding to through and the natural complex structure on obtain a complex structure . It is known that the almost complex on structure on thus obtained is integrable (Salamon [8]). Since is invariant under the action of $c*$ , it can be pushed down to a globally defined integrable becomes almost complex structure on $Z=P(F)$ , in such a way that $c*$ a (locally defined) holomorphic is a complex sub-bundle and that each manifold biholomorphic to of type $(1, 0)$ . Note that the space forms on is given by $\hat{\mathcal{V}}$
$\hat{\mathcal{H}}$
$\hat{\mathcal{H}}_{h}$
$h\in F_{0}$
$p_{*}$
$\hat{\mathcal{V}}_{h}=T_{h}(F_{0})_{x}\cong F_{x}$
$\hat{\mathcal{H}}_{h}$
$\hat{\mathcal{V}}_{h}$
$\pi(h)$
$p_{*}$
$\hat{J}_{h}$
$\hat{J}$
$T_{h}F_{0}$
$\hat{J}$
$F_{0}$
$\pi:F_{0}arrow Z$
$\overline{J}$
$Z_{x}$
$\wedge^{1.0}(\hat{\mathcal{H}}_{h})$
$P_{1}(C)$
$\hat{\mathcal{H}}_{h}$
(2.8)
$\wedge^{1.0}(\hat{\mathcal{H}}_{h})=p^{*}(E_{x}\otimes Ch)$
The antilinear map of it induces an antiholomorphic as is nothing but called the canonical involution morphic automorphisms of $Z$ , $harrow\tilde{h}$
$Z\subset \mathcal{I}mH,$
$\tau$
.
. Therefore for each satisfies $Z$ $Z$ with is regarded . If involution of . It is the antipodal map on each fibre of $Z$ . Let $Aut(Z)$ denote the group of all holoand put $h\cdot a=\tilde{h}\cdot\overline{a}\sim$
$a\in C^{*}$
$F_{0}$
$\tau$
$q\circ\tau=q$
$Aut(Z, \tau)=\{\phi\in Aut(Z) ; \phi\tau=\tau\phi\}$
$Z_{x}\cong S^{2}$
.
150
T. NITTA and M. TAKEUCHI
For
$\psi\in Aut(M, g, H)$
we define the lift
$\overline{\psi}(z)=\psi_{*}z\psi_{*}^{-1}$
for
$\overline{\psi}$
of
$\psi$
by
$z\in Z\subset \mathcal{I}mH\subset End(TM)$
.
Then belongs to $Aut(Z, \tau)$ and satisfies , and moreover leaves $TZ$ invariant the horizontal distribution in . The homomorphism $Aut(M, g, H)$ $arrow-Aut(Z, \tau)$ is injective and called the lifting. $\overline{\psi}$
$q\circ\overline{\psi}=\overline{\psi}^{\circ}q$
$\overline{\psi}*$
$\mathcal{H}$
Let
denote the (locally defined) tautological holomorphic line bundle over $Z=P(F)$ . Then is a globally defined holomorphic line bundle over . Let and define $L^{-1}$
$Z$
$L^{2}$
$Z_{2}=\{\pm 1I\subset C^{*}$
$P=F_{0}/Z_{2}$
,
which is a globally defined complex manifold. Let : jection induced by . Denoting by $\{h\}\in P$ the class of holomorphic action of on by
$h\in F_{0}$
$\pi$
denote the pro, we define a free
$Parrow Z$
$\hat{\pi}$
$P$
$C^{*}$
for
$\{h\}\cdot a=\{h\cdot a^{-1/2}\}$
$a\in C^{*}$
$h\in F_{0},$
.
. In fact, associated to with $z=\pi(h)$ , let $\xi\in(Ch)^{*}=L_{z}$ satisfy $\xi(h)=1$ . Then the correspondgives the required holomorphic -bundle isomorphism $Parrow P(L^{2})$ . ence by Next we define a (locally defined) smooth C-valued l-form on Then for
$\hat{\pi}$
:
is identified with the
$Parrow Z$
$C^{*}$
-bundle
$L^{2}$
$P(L^{2})$
$h\in F_{0}$
$C^{*}$
$\{h\}rightarrow\xi\otimes\xi$
$F_{0}$
$\theta$
$\theta(X)=\omega_{F}(c_{V}(X), h)$
:
for
$X\in T_{h}F_{0},$
$h\in F_{0}$
,
is the connection map for the induced connection on , and hence it is pushed on . It is invariant under the action of . , down to a globally dePned l-form on . Then is holomorphic on it is a type $(1, 0)$ form with $d’’\theta=0$ (Salamon [8], p. 154) and satisfies (1.4), (1.5). Furthermore, if the scalar curvature satisfies of $(M, g)$ is not zero, also (1.6). This follows from the formula (cf. Salamon [8], p. 155)
where
$c_{\nabla}$
$T_{h}Farrow F_{x},$
$x=p(h)$ ,
$F$
$\nabla$
$F_{0}$
$Z_{2}$
$\theta$
$P$
$P,$
$\theta$
$e.$
$\theta$
$t$
(2.9)
$i$
$\Theta=d\theta=-2\nu P^{*}(\omega_{E}\otimes h\otimes h)-2dz^{1}\wedge dz^{Z}$
at
$h\in F_{0}$
,
is the fibre coordinate of $F$ with respect to a where $\nu=t/8n(n+2)$ and of $F$ around $x=P(h)$ with $(Vh_{i})_{x}=0$ . Therefore, local standard basis , by Theorem 1.4 defines a complex contact structure 7 on $Z$ with if by definition of . It is called the canonical . Note that we have $Z$ complex contact structure on . We put $(z^{1}, z^{2})$
$\{h_{1}, h_{2}\}$
$t\neq 0$
$L_{\gamma}\cong L^{2}$
$\theta$
$\theta$
$E_{\gamma}=\mathcal{H}$
$Aut(Z, \gamma, \tau)=Aut(Z, \gamma)\cap Aut(Z, \tau)$
.
Then the lifting is an injective homomorphism $Aut(M, g, H)arrow Aut(Z, \gamma, \tau)$ , since invariant. the lift of $\psi\in Aut(M, g, H)$ leaves $(M, g)$ Suppose again that has non-zero scalar curvature . The hermitian $\mathcal{H}=E_{\gamma}$
$t$
Twistor spaces
151
fibre metric on induced by the metric defined hermitian fibre metric on . We put $\overline{\omega}_{0}=-\frac{\sqrt{-1}}{2}$
which is a real closed 2-form on
2-tensor
on
$Z$
(curvature
form of
of type
,
$Z$
if
(2.11)
$a$
),
and then define a symmetric
.
$Y\in T_{z}Z$
$X,$
we normalize it as
We have then $t>0$
$(1, 1)$
for
$\overline{g}=\frac{1}{\nu}\overline{g}_{0}$
(2.10)
determines a globally
by $\overline{g}_{0}(X, Y)=\overline{\omega}_{0}(\overline{J}X, Y)$
Finally
$F$
$L^{2}$
$a$
$\overline{g}_{0}$
on
$\omega_{F}(h,\tilde{h})$
$L^{-1}$
$\overline{g}$
$\overline{g}$
(Salamon [8])
is a pseudo-K\"ahler metric on
(resp.
.
$Z$
of signature
$(2n+1,0)$ (resp.
$(2n,$
$1)$
)
$t0$ (resp. $t
